Lower limit topology: Difference between revisions

Latest revision as of 01:30, 25 September 2014

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First wave of Yahoo and google Chromebooks arrives

BEAUFORT The first shipment associated with Google Chromebooks has arrived and will be engaging in the hands of county students and teachers during the next two weeks.

When he sorted boxes of Chromebooks Saturday at a school system storage space warehouse, Todd Williamson, technology representative for the county school technique, said 900 of the Yahoo and google Chrome OS operating systems entered Jan. 2, and a second shipment of 960 will arrive in the future this winter.

The purchase of your computers was part of a 3 year technology plan. The initial goal was to provide a Chromebook for those students in grades three through 12 when the program was fully implemented.

Having said that, Mr. Williamson said he is when developing a new four Nike Cortez Nylon White year engineering replacement plan that includes when buying Chromebooks or "similar devices" for all students in grades kindergarten through Twelve. He plans to present which plan during the upcoming Nation Board of Education budget tells you for fiscal 2014 15. Which plan would replace the unique three year plan.

They wasn't yet ready to reveal the estimated cost with the new plan. He said he'd have a figure ready to given to the board members once they begin budget workshops, which have been scheduled to begin later the following month.

Two reasons to increase the technology plan are to replace outdated equipment and to help offset cuts to state publication funds, which has been slashed 80 % over the last four years.

"When we're not receiving textbook money from the express, we must increase access to Over the internet resources," he said.

Classes districts are expected to make the switch to digital textbooks by 2017 Fake Longchamp Bags Uk despite the fact that no state money is set aside for computers or other electronic devices for every student.

Physician. Heather Dietzler, director of federal government programs and grants for the county school system, stated Tuesday during a County Plank of Education meeting that providing the Chromebooks to each student is also critical because all state examining will be done online in the future.

An original three year plan called for the purchase of about 6,1000 Chromebooks for students in grades a number of through 12. He said Nike Blazers Low some year plan calls for the purchase of about 8,500 Chromebooks or very similar devices for students in qualities kindergarten through 12.

When he plans to stay with Chromebooks for degrees four through 12, he was quoted saying they might choose other systems for the early grades.

"Their wants may require something a little different,In he said.

This year's shipment of Chromebooks is being purchased with $350,000 in Race to the Top federal grant funds, provided for schooling improvement initiatives, and $330,1000 in 2013 14 local capital funds.

The first samsung wave s8500 of books that just came was purchased with county capital money, with the second wave later this winter coming from Race to the Top revenue. The County Board of Education awarded a $344,347 contract The following friday to Technology Integration Set, with offices in several big cities, including Richmond, Va., because vendor for the second transport. The first shipment was by means of CDWG, a group based in Connecticut.

Course instructors had to submit proposals at the end of the 2012 13 college year on how they would take advantage of the Chromebooks to qualify for devices from this year's shipments. A course and instruction team, of which Mr. Williamson was a member, picked the winners. The team been given about 85 proposals, by using 70 teachers selected.

Mr. Williamson and Wes Rinehart, director of engineering support services, distributed books to Beaufort Middle School, Beaufort Elementary plus East Carteret High School today and is handing out books next week to help teachers at seven universities. They are: Morehead City Middle School, Bogue Sound Elementary School, Croatan High School, Large Creek Middle School, White Cherry Elementary School, Newport Middle School as well as West Carteret High School. Other institutions will receive books in the next shipping.

"Wes and I are going to be distributing them to the schools this week and next full week so that teachers can begin usage of them to start second semester classes (which begin Present cards. 22)," said Mister. Williamson. "Once they are distributed to schools, they are going to need to be inventoried and then installed in the carts for delivery towards teachers."

Superintendent Dr. David Novey said the purchase of so many Chromebooks at one time would not have been possible minus the support of county commissioners and also Race to the Top funds.

The teachers district received $892,000 in Race to the Top federal allow funds four years ago to fund eight education improvement campaigns. School officials decided to use $350,000 of that for the Chromebooks.

Competition to the Top is a $4 billion federal education reform initiative which has awarded grants to states over the last four years to implement schooling reforms and improvements. States that went for grants agreed to use a part of the money to implement federal Common Core standards, which outlines national reading and calculations standards that students should meet. As of December 2013, 45 states have adopted the standards, including North Carolina.

States can apply for a portion of the $4 billion. Idaho received $400 million four years earlier, and in turn awarded funds so that you can individual school districts. Priority was given to districts which has a high percentage of low educational performing students and a great poverty rate. Carteret doesn't gain a high position in either category.

The state offer is scheduled to expire no more this school year, although state education leaders usually are asking for an extension of the give period.

Dr. Dietzler said a district used Race to the top level funds for eight schooling improvement efforts.

They consist of: the purchase of the Chromebooks; staff growth for implementation of the Common Core standards; technology tools and training; data methods training; creating assessments; setting up a transition plan and boosting infrastructure Buy Air Jordans Online for Home Bottom, a statewide website that households student and teacher data and provides a portal for parents to access their child's qualities.

Other funds have been familiar with train teachers and directors in the use of the Education Useful Assessment System, known as EVAAS, a data system that analyzes scholar academic growth; and giving administrators to leadership teaching academies.

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Aaron Hernandez latest proof New England Patriots have lost their 'Way'

things that people refer to as intangibles we do not see as intangible in any respect, Pioli told the media prior to Completely new England's 32 29 victory more than Carolina in Super Bowl XXXVIII, last 2004.

player makeup, her character, integrity, leadership all those things are very tangible. And the've a say in how you would try to build our hockey team. a philosophy turned known as the Patriot Way.

The latest reason to question some of the personnel judgements made in recent years by go coach Bill Belichick: Tight conclude Aaron Hernandez potential involvement in a the latest Massachusetts homicide case.

Not that Belichick or the Patriots need apologize for 1 of the most impressive stretches within NFL history. The results communicate for themselves.

New England has picked up at least 10 games pertaining to 10 consecutive seasons. Along the way, the Patriots have played in five Super Bowls and picked up three of them. They achieved two other AFC Championship Games, including last time 28 13 home burning to eventual Super Bowl champion Baltimore.

Yet there was a time when it seemed New England owner Robert Kraft might need to build an extension upon Patriot Place to fit more Lombardi Awards. Instead, six other franchise companies the New York Giants (two times), Pittsburgh (twice), Baltimore, Green Bay, Brand-new Orleans and Indianapolis have captured titles since the Patriots defeated Philly in Super Bowl XXXIX within 2005.

There is no single aspect to explain why the Patriots have fallen short. Some years there was better teams. Tom Brady missed almost the entire 2008 year or so with a knee injury. Legend tight end Rob Gronkowski was gimpy in Super Bowl XLVI.

Genuine luck plays a role, as well. With a couple of breaks here and there, Brady could have enough rings for hands.

One of the reasons he doesn The quality of Brady supporting cast has slowly but surely diminished. His quarterbacking prowess along with the outstanding coaching of Belichick and his awesome staff has largely helped the Patriots overcome the drop in talent.

Trying to compensate for of which dip, Belichick has proved extra willing in recent years to gamble on players with character issues after finding before success with wide phone Randy Moss and running back Corey Dillon.

Obviously, not every NFL player can have the choir boy image of Ricky Tebow (who, ironically, may end way up helping the Patriots weather the likely loss of Buy Mont Blanc Pen Online Uk Hernandez if Tebow converts through quarterback to tight finish). But Belichick Cheap Bailey Button Boots belief that some leopards may change their spots has created its set of headaches and negative publicity even as the majority of Patriots have steered clear of legal trouble and bad decisions.

The Patriots dealt out draft choices for wide phone Chad Johnson and defensive take on Albert Haynesworth in 2011 after inappropriate behavior with their New Balance Mt110 prior teams received them placed on the investing block. Neither caused issues in New England, but both were major flops.

New England created cornerback Alfonzo Dennard Ralph Lauren Polo Shirts Uk a 2012 seventh round choice despite a predraft offender arrest on charges they assaulted a police officer. Dennard recently was sentenced to 30 days in jail adopting the 2013 season and two decades probation.

The Patriots traded with Polk Bay last October pertaining to cornerback Aqib Talib, who served a three game suspension earlier while in the season for a PED violation. Talib served a one game suspension this season for violating the Nhl personal conduct policy following being arrested for simple battery pack and resisting arrest the first sort season.

Tampa Bay as well sent running back LaGarrette Blount to New England in April. Blount offered a 10 game suspension while at the University of Modifies name after sucker punching your Boise State football player with a backlash.

The current situation with Hernandez stings essentially the most. The Patriots were so assured Hernandez had moved beyond the college or university character concerns that brought on his fourth round glide in the 2010 draft which he was given a seven calendar year, $41.2 million contract extension with a $12.5 million signing bonus during the 2012 offseason.

As much as Northeastern prosperity during its function of three Super Dish wins bred contempt between opposing teams and fans, the professionalism of Patriots participants was respected. The 2004 roster featured 35 avid gamers with college degrees, the NFL second highest total regarding Carolina (42) and Indianapolis (Thirty eight). The Patriots were considered as smart as they were athletically talented. This showed on the field.

Maybe it was Pioli starting to Kansas City in 2009. There is a mutual respect amongst Pioli and Belichick that generated give to get and sometimes saved the business from taking what after proved to be regrettable risks. Belichick, who may have final say on virtually all personnel moves, might not be obtaining enough pushback from his present-day front office members to stop making these types of mistakes.

Probably Belichick placed too much faith within the predraft feedback of friend Downtown Meyer. Belichick selected five University or college of Florida players from when Meyer was the Gators go coach. Hernandez, Spikes and Cunningham will be three of them.

Wide recipient Chad Jackson, a 2006 second round pick, was a bust. Belichick also cut 2006 in 7th place round pick Jeremy Mincey, who has commenced every game in Jacksonville from defensive end the past a couple of seasons.

Maybe it was just catastrophe and can come with the territory when trying to nab a bargain.

There is absolutely no guessing about what must can come next in New England. Belichick ought to take a long look at the varieties of people he is bringing in to the franchise. The integrity this Pioli spoke about must will carry more weight again.

Greater character on the roster doesn guarantee Belichick and Brady will win an additional Super Bowl before these people retire. But with what is transpiring around New England, it sure wouldn damage.

@@ Line 1: / Line 1: @@
-{{About|equalisers in mathematics|other meanings|Equalizer (disambiguation)}}
+== Croatan High School Nike Cortez Nylon White ==
-In [[mathematics]], an '''equaliser''', or '''equalizer''', is a set of arguments where two or more [[function (mathematics)|function]]s have [[equality (math)|equal]] values.
-An equaliser is the [[solution set]] of an [[equation]].
-In certain contexts, a '''difference kernel''' is the equaliser of exactly two functions.
-== Definitions ==
+First wave of Yahoo and google Chromebooks arrives<br><br>BEAUFORT   The first shipment associated with Google Chromebooks has arrived and will be engaging in the hands of county students and teachers during the next two weeks.<br><br>When he sorted boxes of Chromebooks Saturday at a school system storage space warehouse, Todd Williamson, technology representative for the county school technique, said 900 of the Yahoo and google Chrome OS operating systems entered Jan. 2, and a second shipment of 960 will arrive in the future this winter.<br><br>The purchase of your computers was part of a 3 year technology plan. The initial goal was to provide a Chromebook for those students in grades three through 12 when the program was fully implemented.<br><br>Having said that, Mr. Williamson said he is when developing a new four [http://www.lucyrichardson.co.uk/news/?id=33 Nike Cortez Nylon White] year engineering replacement plan that includes when buying Chromebooks or "similar devices" for all students in grades kindergarten through Twelve. He plans to present which plan during the upcoming Nation Board of Education budget tells you for fiscal 2014 15. Which plan would replace the unique three year plan.<br><br>They wasn't yet ready to reveal the estimated cost with the new plan. He said he'd have a figure ready to given to the board members once they begin budget workshops, which have been scheduled to begin later the following month.<br><br>Two reasons to increase the technology plan are to replace outdated equipment and to help offset cuts to state publication funds, which has been slashed 80 % over the last four years.<br><br>"When we're not receiving textbook money from the express, we must increase access to Over the internet resources," he said.<br><br>Classes districts are expected to make the switch to digital textbooks by 2017 [http://www.lakesview.co.uk/includes/?page=20 Fake Longchamp Bags Uk] despite the fact that no state money is set aside for computers or other electronic devices for every student.<br><br>Physician. Heather Dietzler, director of federal government programs and grants for the county school system, stated Tuesday during a County Plank of Education meeting that providing the Chromebooks to each student is also critical because all state examining will be done online in the future.<br><br>An original three year plan called for the purchase of about 6,1000 Chromebooks for students in grades a number of through 12. He said [http://www.bristolcar-vanhire.co.uk/transfer/?h=64 Nike Blazers Low] some year plan calls for the purchase of about 8,500 Chromebooks or very similar devices for students in qualities kindergarten through 12.<br><br>When he plans to stay with Chromebooks for degrees four through 12, he was quoted saying they might choose other systems for the early grades.<br><br>"Their wants may require something a little different,In he said.<br><br>This year's shipment of Chromebooks is being purchased with $350,000 in Race to the Top federal grant funds, provided for schooling improvement initiatives, and $330,1000 in 2013 14 local capital funds.<br><br>The first samsung wave s8500 of books that just came was purchased with county capital money, with the second wave later this winter coming from Race to the Top revenue. The County Board of Education awarded a $344,347 contract The following friday to Technology Integration Set, with offices in several big cities, including Richmond, Va., because vendor for the second transport. The first shipment was by means of CDWG, a group based in Connecticut.<br><br>Course instructors had to submit proposals at the end of the 2012 13 college year on how they would take advantage of the Chromebooks to qualify for devices from this year's shipments. A course and instruction team, of which Mr. Williamson was a member, picked the winners. The team been given about 85 proposals, by using 70 teachers selected.<br><br>Mr. Williamson and Wes Rinehart, director of engineering support services, distributed books to Beaufort Middle School, Beaufort Elementary plus East Carteret High School today and is handing out books next week to help teachers at seven universities. They are: Morehead City Middle School, Bogue Sound Elementary School, Croatan High School, Large Creek Middle School, White Cherry Elementary School, Newport Middle School as well as West Carteret High School. Other institutions will receive books in the next shipping.<br><br>"Wes and I are going to be distributing them to the schools this week and next full week so that teachers can begin usage of them to start second semester classes (which begin Present cards. 22)," said Mister. Williamson. "Once they are distributed to schools, they are going to need to be inventoried and then installed in the carts for delivery towards teachers."<br><br>Superintendent Dr. David Novey said the purchase of so many Chromebooks at one time would not have been possible minus the support of county commissioners and also Race to the Top funds.<br><br>The teachers district received $892,000 in Race to the Top federal allow funds four years ago to fund eight education improvement campaigns. School officials decided to use $350,000 of that for the Chromebooks.<br><br>Competition to the Top is a $4 billion federal education reform initiative which has awarded grants to states over the last four years to implement schooling reforms and improvements. States that went for grants agreed to use a part of the money to implement federal Common Core standards, which outlines national reading and calculations standards that students should meet. As of December 2013, 45 states have adopted the standards, including North Carolina.<br><br>States can apply for a portion of the $4 billion. Idaho received $400 million four years earlier, and in turn awarded funds so that you can individual school districts. Priority was given to districts which has a high percentage of low educational performing students and a great poverty rate. Carteret doesn't gain a high position in either category.<br><br>The state offer is scheduled to expire no more this school year, although state education leaders usually are asking for an extension of the give period.<br><br>Dr. Dietzler said a district used Race to the top level funds for eight schooling improvement efforts.<br><br>They consist of: the purchase of the Chromebooks; staff growth for implementation of the Common Core standards; technology tools and training; data methods training; creating assessments; setting up a transition plan and boosting infrastructure [http://www.jademetals.co.uk/downloads/?count=78 Buy Air Jordans Online] for Home Bottom, a statewide website that households student and teacher data and provides a portal for parents to access their child's qualities.<br><br>Other funds have been familiar with train teachers and directors in the use of the Education Useful Assessment System, known as EVAAS, a data system that analyzes scholar academic growth; and giving administrators to leadership teaching academies.<ul>
+   <li>[http://www.xjxg.cn/thread-1569969-1-1.html http://www.xjxg.cn/thread-1569969-1-1.html]</li>
+   <li>[http://verdamilio.net/tonio/spip.php?article1544/ http://verdamilio.net/tonio/spip.php?article1544/]</li>
+   <li>[http://general.assembly.codesria.org/spip.php?article87&lang=pt/&quot;/ http://general.assembly.codesria.org/spip.php?article87&lang=pt/&quot;/]</li>
+   <li>[http://enseignement-lsf.com/spip.php?article66#forum32042582 http://enseignement-lsf.com/spip.php?article66#forum32042582]</li>
+ </ul>
-Let ''X'' and ''Y'' be [[Set (mathematics)|sets]].
+== and Indianapolis  Buy Mont Blanc Pen Online Uk ==
-Let ''f'' and ''g'' be [[function (mathematics)|function]]s, both from ''X'' to ''Y''.
-Then the ''equaliser'' of ''f'' and ''g'' is the set of elements ''x'' of ''X'' such that ''f''(''x'') equals ''g''(''x'') in ''Y''.
-Symbolically:
-: <math> \mathrm{Eq}(f,g) := \{x \in X \mid f(x) = g(x)\}\mbox{.}\! </math>
-The equaliser may be denoted Eq(''f'',''g'') or a variation on that theme (such as with lowercase letters "eq").
-In informal contexts, the notation {''f'' = ''g''} is common.
-The definition above used two functions ''f'' and ''g'', but there is no need to restrict to only two functions, or even to only [[finite set|finite]]ly many functions.
+Aaron Hernandez latest proof New England Patriots have lost their 'Way'<br><br>things that people refer to as intangibles we do not see as intangible in any respect, Pioli told the media prior to Completely new England's 32 29 victory more than Carolina in Super Bowl XXXVIII, last 2004.<br><br>player makeup, her character, integrity, leadership all those things are very tangible. And the've a say in how you would try to build our hockey team. a philosophy turned known as the Patriot Way.<br><br>The latest reason to question some of the personnel judgements made in recent years by go coach Bill Belichick: Tight conclude Aaron Hernandez potential involvement in a the latest Massachusetts homicide case.<br><br>Not that Belichick or the Patriots need apologize for 1 of the most impressive stretches within NFL history. The results communicate for themselves.<br><br>New England has picked up at least 10 games pertaining to 10 consecutive seasons. Along the way, the Patriots have played in five Super Bowls and picked up three of them. They achieved two other AFC Championship Games, including last time 28 13 home burning to eventual Super Bowl champion Baltimore.<br><br>Yet there was a time when it seemed New England owner Robert Kraft might need to build an extension upon Patriot Place to fit more Lombardi Awards. Instead, six other franchise companies the New York Giants (two times), Pittsburgh (twice), Baltimore, Green Bay, Brand-new Orleans and Indianapolis have captured titles since the Patriots defeated Philly in Super Bowl XXXIX within 2005.<br><br>There is no single aspect to explain why the Patriots have fallen short. Some years there was better teams. Tom Brady missed almost the entire 2008 year or so with a knee injury. Legend tight end Rob Gronkowski was gimpy in Super Bowl XLVI.<br><br>Genuine luck plays a role, as well. With a couple of breaks here and there, Brady could have enough rings for hands.<br><br>One of the reasons he doesn The quality of Brady supporting cast has slowly but surely diminished. His quarterbacking prowess along with the outstanding coaching of Belichick and his awesome staff has largely helped the Patriots overcome the drop in talent.<br><br>Trying to compensate for of which dip, Belichick has proved extra willing in recent years to gamble on players with character issues after finding before success with wide phone Randy Moss and running back Corey Dillon.<br><br>Obviously, not every NFL player can have the choir boy image of Ricky Tebow (who, ironically, may end way up helping the Patriots weather the likely loss of [http://www.pdsconsultants.co.uk/skins/?id=47 Buy Mont Blanc Pen Online Uk] Hernandez if Tebow converts through quarterback to tight finish). But Belichick [http://www.adoptadolphin.org.uk/content/?p=22 Cheap Bailey Button Boots] belief that some leopards may change their spots has created its set of headaches and negative publicity even as the majority of Patriots have steered clear of legal trouble and bad decisions.<br><br> The Patriots dealt out draft choices for wide phone Chad Johnson and defensive take on Albert Haynesworth in 2011 after inappropriate behavior with their [http://www.elastomerics.com/js/?id=95 New Balance Mt110] prior teams received them placed on the investing block. Neither caused issues in New England, but both were major flops.<br><br> New England created cornerback Alfonzo Dennard [http://www.jonahoad.com/email/?page=9 Ralph Lauren Polo Shirts Uk] a 2012 seventh round choice despite a predraft offender arrest on charges they assaulted a police officer. Dennard recently was sentenced to 30 days in jail adopting the 2013 season and two decades probation.<br><br> The Patriots traded with Polk Bay last October pertaining to cornerback Aqib Talib, who served a three game suspension earlier while in the season for a PED violation. Talib served a one game suspension this season for violating the Nhl personal conduct policy following being arrested for simple battery pack and resisting arrest the first sort season.<br><br> Tampa Bay as well sent running back LaGarrette Blount to New England in April. Blount offered a 10 game suspension while at the University of Modifies name after sucker punching your Boise State football player with a backlash.<br><br>The current situation with Hernandez stings essentially the most. The Patriots were so assured Hernandez had moved beyond the college or university character concerns that brought on his fourth round glide in the 2010 draft which he was given a seven calendar year, $41.2 million contract extension with a $12.5 million signing bonus during the 2012 offseason.<br><br>As much as Northeastern prosperity during its function of three Super Dish wins bred contempt between opposing teams and fans, the professionalism of Patriots participants was respected. The 2004 roster featured 35 avid gamers with college degrees, the NFL second highest total regarding Carolina (42) and Indianapolis (Thirty eight). The Patriots were considered as smart as they were athletically talented. This showed on the field.<br><br>Maybe it was Pioli starting to Kansas City in 2009. There is a mutual respect amongst Pioli and Belichick that generated give to get and sometimes saved the business from taking what after proved to be regrettable risks. Belichick, who may have final say on virtually all personnel moves, might not be obtaining enough pushback from his present-day front office members to stop making these types of mistakes.<br><br>Probably Belichick placed too much faith within the predraft feedback of friend Downtown Meyer. Belichick selected five University or college of Florida players from when Meyer was the Gators go coach. Hernandez, Spikes and Cunningham will be three of them.<br><br>Wide recipient Chad Jackson, a 2006 second round pick, was a bust. Belichick also cut 2006 in 7th place round pick Jeremy Mincey, who has commenced every game in Jacksonville from defensive end the past a couple of seasons.<br><br>Maybe it was just catastrophe and can come with the territory when trying to nab a bargain.<br><br>There is absolutely no guessing about what must can come next in New England. Belichick ought to take a long look at the varieties of people he is bringing in to the franchise. The integrity this Pioli spoke about must will carry more weight again.<br><br>Greater character on the roster doesn guarantee Belichick and Brady will win an additional Super Bowl before these people retire. But with what is transpiring around New England, it sure wouldn damage.<ul>
-In general, if '''F''' is a [[Set (mathematics)|set]] of functions from ''X'' to ''Y'', then the ''equaliser'' of the members of '''F''' is the set of elements ''x'' of ''X'' such that, given any two members ''f'' and ''g'' of '''F''', ''f''(''x'') equals ''g''(''x'') in ''Y''.
-Symbolically:
+   <li>[http://verdamilio.net/tonio/spip.php?article1739/ http://verdamilio.net/tonio/spip.php?article1739/]</li>
-: <math> \mathrm{Eq}(\mathcal{F}) := \{x \in X \mid \forall{f,g \,}{\in}\, \mathcal{F}, \; f(x) = g(x)\}\mbox{.}\! </math>
-This equaliser may be written as Eq(''f'',''g'',''h'',...) if <math> \mathcal{F}</math> is the set {''f'',''g'',''h'',...}.
+   <li>[http://bbs.dashunde.com/viewthread.php?tid=340146&extra= http://bbs.dashunde.com/viewthread.php?tid=340146&extra=]</li>
-In the latter case, one may also find {''f'' = ''g'' = ''h'' = ···} in informal contexts.
+   <li>[http://www.shaffaf.net/spip.php?article929&lang=ar&id_forum=27372/ http://www.shaffaf.net/spip.php?article929&lang=ar&id_forum=27372/]</li>
-As a [[degenerate (math)|degenerate]] case of the general definition, let '''F''' be a [[singleton (set theory)|singleton]] {''f''}.
-Since ''f''(''x'') always equals itself, the equaliser must be the entire domain ''X''.
+   <li>[http://verdamilio.net/tonio/spip.php?article1970/ http://verdamilio.net/tonio/spip.php?article1970/]</li>
-As an even more degenerate case, let '''F''' be the [[empty set]] {}.
-Then the equaliser is again the entire domain ''X'', since the [[universal quantification]] in the definition is [[vacuously true]].
+ </ul>
-== Difference kernels ==
-A binary equaliser (that is, an equaliser of just two functions) is also called a ''difference kernel''.
-This may also be denoted DiffKer(''f'',''g''), Ker(''f'',''g''), or Ker(''f'' &minus; ''g'').
-The last notation shows where this terminology comes from, and why it is most common in the context of [[abstract algebra]]:
-The difference kernel of ''f'' and ''g'' is simply the [[kernel (algebra)|kernel]] of the difference ''f'' &minus; ''g''.
-Furthermore, the kernel of a single function ''f'' can be reconstructed as the difference kernel Eq(''f'',0), where 0 is the [[constant function]] with value [[0 (number)|zero]].
-Of course, all of this presumes an algebraic context where the kernel of a function is its [[preimage]] under zero; that is not true in all situations.
-However, the terminology "difference kernel" has no other meaning.
-== In category theory ==
-Equalisers can be defined by a [[universal property]], which allows the notion to be generalised from the [[category of sets]] to arbitrary [[category theory|categories]].
-In the general context, ''X'' and ''Y'' are objects, while ''f'' and ''g'' are morphisms from ''X'' to ''Y''.
-These objects and morphisms form a [[commutative diagram|diagram]] in the category in question, and the equaliser is simply the [[limit (category theory)|limit]] of that diagram.
-In more explicit terms, the equaliser consists of an object ''E'' and a morphism ''eq'' : ''E'' → ''X'' satisfying <math>f \circ eq = g \circ eq</math>,
-and such that, given any object ''O'' and morphism ''m'' : ''O'' → ''X'', if <math>f \circ m = g \circ m</math>, then there exists a [[unique]] morphism ''u'' : ''O'' → ''E'' such that <math>eq \circ u = m</math>.
-<div style="text-align: center;">[[Image:Equalizer-01.svg|200px]]</div>
-A morphism <math>m:O \rightarrow X</math> is said to '''equalize''' <math>f</math> and <math>g</math> if <math>f \circ m = g \circ m</math>.
-<ref>{{cite book |last1=Barr |first1=Michael |authorlink1=Michael Barr (mathematician) |last2=Wells |first2=Charles |authorlink2=Charles Wells (mathematician) |year=1998 |title=Category theory for computing science |page=266 |url=http://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf |accessdate=2013-07-20 |format=PDF}}</ref>
-In any [[universal algebra]]ic category, including the categories where difference kernels are used, as well as the category of sets itself, the object ''E'' can always be taken to be the ordinary notion of equaliser, and the morphism ''eq'' can in that case be taken to be the [[inclusion function]] of ''E'' as a [[subset]] of ''X''.
-The generalisation of this to more than two morphisms is straightforward; simply use a larger diagram with more morphisms in it.
-The degenerate case of only one morphism is also straightforward; then ''eq'' can be any [[isomorphism]] from an object ''E'' to ''X''.
-The correct diagram for the degenerate case with ''no'' morphisms is slightly subtle: one might initially draw the diagram as consisting of the objects ''X'' and ''Y'' and no morphisms. This is incorrect, however, since the limit of such a diagram is the [[product (category theory)|product]] of ''X'' and ''Y'', rather than the equalizer. (And indeed products and equalizers are different concepts: the set-theoretic definition of product doesn't agree with the set-theoretic definition of the equalizer mentioned above, hence they are actually different.) Instead, the appropriate insight is that every equalizer diagram is fundamentally concerned with ''X'', including ''Y'' only because ''Y'' is the [[codomain]] of morphisms which appear in the diagram. With this view, we see that if there are no morphisms involved, ''Y'' does not make an appearance and the equalizer diagram consists of ''X'' alone. The limit of this diagram is then any isomorphism between ''E'' and ''X''.
-It can be proved that any equaliser in any category is a [[monomorphism]].
-If the [[converse (logic)|converse]] holds in a given category, then that category is said to be ''regular'' (in the sense of monomorphisms).
-More generally, a [[regular monomorphism]] in any category is any morphism ''m'' that is an equaliser of some set of morphisms.
-Some authorities require (more strictly) that ''m'' be a ''binary'' equaliser, that is an equaliser of exactly two morphisms.
-However, if the category in question is [[complete category|complete]], then both definitions agree.
-The notion of difference kernel also makes sense in a category-theoretic context.
-The terminology "difference kernel" is common throughout category theory for any binary equaliser.
-In the case of a [[preadditive category]] (a category [[enriched category|enriched]] over the category of [[Abelian group]]s), the term "difference kernel" may be interpreted literally, since subtraction of morphisms makes sense.
-That is, Eq(''f'',''g'') = Ker(''f'' - ''g''), where Ker denotes the [[kernel (category theory)|category-theoretic kernel]].
-Any category with fibre products (pull backs) and products has equalisers.
-== See also ==
-*[[Coequaliser]], the [[dual (category theory)|dual]] notion, obtained by reversing the arrows in the equaliser definition.
-*[[Coincidence theory]], a topological approach to equalizer sets in [[topological space]]s.
-*[[Pullback (category theory)|Pullback]], a special [[Limit (category theory)|limit]] that can be constructed from equalisers and products.
-==Notes==
-{{reflist}}
-==References==
-* {{nlab|id=equalizer|title=Equalizer}}
-== External links ==
-*[http://www.j-paine.org/cgi-bin/webcats/webcats.php Interactive Web page ] which generates examples of equalizers in the category of finite sets. Written by [http://www.j-paine.org/ Jocelyn Paine].
-[[Category:Set theory]]
-[[Category:Limits (category theory)]]

Lower limit topology: Difference between revisions

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